Inexact Methods for Black-Oil Sequential Fully Implicit SFI Scheme

Abstract

The sequential fully implicit (SFI) scheme was introduced (Jenny et al. 2006) for solving coupled flow and transport problems. Each time step for SFI consists of an outer loop, in which there are inner Newton loops to implicitly and sequentially solve the pressure and transport sub-problems. In standard SFI, the sub-problems are usually fully solved at each outer iteration. This can result in wasted computations that contribute little towards the coupled solution. The issue is known as ‘over-solving’. Our objective is to minimize the cost while maintain or improve the convergence of SFI by preventing ‘over-solving’. We first developed a framework based on the nonlinear acceleration techniques (Jiang and Tchelepi 2019) to ensure robust outer-loop convergence. We then developed inexact-type methods that prevent ‘over-solving’ and minimize the cost of inner solvers for SFI. The motivation is similar to the inexact Newton method, where the inner (linear) iterations are controlled in a way that the outer (Newton) convergence is not degraded, but the overall computational effort is greatly reduced. We proposed an adaptive strategy that provides relative tolerances based on the convergence rates of the coupled problem. The developed inexact SFI method was tested using numerous simulation studies. We compared different strategies such as fixed relaxations on absolute and relative tolerances for the inner solvers. The test cases included synthetic as well as real-field models with complex flow physics and high heterogeneity. The results show that the basic SFI method is quite inefficient. When the coupling is strong, we observed that the outer convergence is mainly restricted by the initial residuals of the sub-problems. It was observed that the feedback from one inner solver can cause the residual of the other to rebound to a much higher level. Away from a coupled solution, additional accuracy achieved in inner solvers is wasted, contributing to little or no reduction of the overall residual. By comparison, the inexact SFI method adaptively provided the relative tolerances adequate for the sub-problems. We show across a wide range of flow conditions that the inexact SFI can effectively resolve the ‘over-solving’ issue, and thus greatly improve the overall performance. The novel information of this paper includes: 1) we found that for SFI, there is no need for one sub-problem to strive for perfection (‘over-solving’), while the coupled residual remains high because of the other sub-problem; 2) a novel inexact SFI method was developed to prevent ‘over-solving’ and minimize the cost of inner solvers; 3) an adaptive strategy was proposed for relative tolerances based on the convergence rates of the coupled problem; and 4) a novel SFI framework was developed based on the nonlinear acceleration techniques to ensure robust outer-loop convergence.